Page 93 - Advanced Linear Algebra

Linear Transformations 77

and is projection onto im²³ along

ker²³ ~ im ² ³

£
)
2 Conversely, if
=~ : l Ä l :
and if is projection onto : along the direct sum £ : ,, then

bÄb ~ is a resolution of the identity.
bÄb ~ is a resolution of the identity, then
Proof. To prove 1), if
=~ im ² ³ b Ä b im ² ³

Moreover, if
%b Ä b ~
%

then applying gives %~ and so the sum is direct. As to the kernel of ,

we have
p s
im²³ l ker ²³ ~ = ~ im²³ l im² ³

q £ t
and since ~ , it follows that

im²³ ‹ ker ² ³

£
and so equality must hold. For part 2), suppose that
=~ : l Ä l :
and is projection onto along £ : . If £ , then
:

im²³ ~ : ‹ ker ² ³

and so ž . Also, if # ~ bÄb for  : , then
# ~ b Äb ~ #bÄb # ~ ² b Äb ³#

and so ~ b b Ä is a resolution of the identity.
The Algebra of Projections

If and are projections, it does not necessarily follow that b , c or
is a projection. For example, the sum b is a projection if and only if

²b ³ ~ b